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Everyone agreed that Sirius was, most of the time, mostly okay. This made it frustrating when he inevitably ended up binge drinking on a weeknight, or walking out of a meeting abruptly, or lost his temper, because nobody could ever be properly furious with him for it. (Worst of all, this included Sirius himself.) It felt as though he could do no wrong simply by virtue of coming back from Azkaban alive and mostly sane.
Never before had Sirius gone a terribly large amount of time without people furious with him for things that were in no way his fault or under his control. This new experience was not pleasant. It felt as though he was walking on clouds and at any minute gravity would reassert itself, and Sirius would go plummeting thousands of feet to splatter on the ground. It was terrifying.
Harry was a relief from that in some ways, because Harry had only a dim notion of what it meant to come back from Azkaban or to be Sirius Black, Third Of His Name. Therefore Harry mostly interacted with Sirius like he did any normal human being, and did not seem either amazed at his civility or categorically disgusted with his existence - both already common reactions before prison. (In retrospect Sirius sometimes had a terrible suspicion about his friendship with James: that it had always been founded on the fact that James was so incredibly self-centered he was oblivious to what everyone else expected of Sirius, and drew so much attention that Sirius attaching to him like an amoeba had turned them into Sirius-and-James instead of Sirius Black, Third Of His Name.)
On the other hand, there was the fact that every adult in Harry's life had monumentally failed him in every fucking way even more comprehensively than the adults in Sirius's life had failed him. Treating Harry with the bare minimal responsibility adults were supposed to show children in their vicinity, let alone their very own bleeding wards, was a revelation to Harry at every turn. This meant that sometimes Sirius had to go off after talking to Harry and scream at Narcissa or Andromeda about how on earth Dumbledore had found someone worse than their parents to raise his godson. (They had both accused him of exaggerating about this until actually meeting Harry for an extended time, at which both of them had separately and spontaneously apologized after.) Also it meant Sirius and Harry hardly had what you would call a normal parental relationship. Sirius had never seen one of these in his life and was well aware of it. Harry, alarmingly, didn't seem to know.
Harry had given Sirius a lifeline in more ways than one. There was Sirius's actual escape, yes, but more than that there was what had come after - after everything. The problem, specifically, was what came after Peter was in prison and Sirius was cleared, and getting from one day to the next was no longer a monumental and odds-defying task, and instead Sirius had to face the thought of living a life he was not obligated to continue. Harry had given him a place to start that was, more or less, manageable. Sirius had always wanted and expected children, insofar as he had expected to make it to the age of twenty-five. (It had taken him a disturbing amount of time to remember who he had expected to have them with, something he was never going to admit had been a happy enough memory to lose. Andromeda had already been disturbed to jar loose the memory of his engagement to Bellatrix with a reminder and realize it had been taken.)
Furthermore, raising children was an intensive task but not at its heart a complicated one. You made sure they had enough to eat and clothing and shelter, and helped if they were sick or scared or unhappy, and you taught them about life, more or less by living it alongside them. Sirius didn't have to be anyone in particular to be Harry Potter's guardian, although the extensive knowledge of Dark Arts and combat magic certainly helped. He could take things one day at a time: answer Harry's letters from hospital. Hire someone to go and clean out the muggle house so that it was fit for humanity. Shout at Dumbledore when he attempted to dissuade Sirius from taking Harry. Go and get Harry from his aunt and uncle. (Do not murder Harry's aunt and uncle.) Grocery shop, shop for bedroom stuff, help with summer homework, ensure there is reasonably nutritious food on the table.
Granted, Sirius had learned some of this stuff in a chaotic mess himself after running away and subsequently purchasing himself a house. (The recent repair people had filled in the holes he hadn't fixed when he was eighteen for ideological reasons, which he was profoundly grateful for and embarrassed about.) But he had more or less had them down by the time he'd been arrested.
This sort of careful lining up of days gave Sirius time, too, to recover his memories and remember how being a person worked, and try to answer some of the letters from old friends as a first step. The responsibility of the Blacks weighed on Sirius like a ball and chain whenever he thought about it, so he tried not to. Arcturus hadn't died so long ago, he reasoned, and the stewards could make it without him another year. (He had letters from the sheriff and various House officials and oddly from Aunt Dru and Irma Crabbe, who he had a foggy recollection appeared on the family tree as Granddad's divorced wife. All of this, he was ignoring.)
Sirius needed time. Dedicating himself wholeheartedly to Harry would give him time. He took on the Defense teaching post Dumbledore offered him, curse and all, with the relief of someone given a bridge out of a crisis. That was eccentric, but it was a basically respectable sort of eccentricity and it would allow him to remember how being in magical society worked. Furthermore, being at Hogwarts would conveniently prevent the press or the Ministry or, Merlin help him, his actual dependents from storming up to him and making him account for himself. Harry was both responsibility and reason to live, all alone.